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Author: John L. Bell Publisher: Springer Nature ISBN: 3030187071 Category : Mathematics Languages : en Pages : 320
Book Description
This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal.
Author: John L. Bell Publisher: Springer Nature ISBN: 3030187071 Category : Mathematics Languages : en Pages : 320
Book Description
This book explores and articulates the concepts of the continuous and the infinitesimal from two points of view: the philosophical and the mathematical. The first section covers the history of these ideas in philosophy. Chapter one, entitled ‘The continuous and the discrete in Ancient Greece, the Orient and the European Middle Ages,’ reviews the work of Plato, Aristotle, Epicurus, and other Ancient Greeks; the elements of early Chinese, Indian and Islamic thought; and early Europeans including Henry of Harclay, Nicholas of Autrecourt, Duns Scotus, William of Ockham, Thomas Bradwardine and Nicolas Oreme. The second chapter of the book covers European thinkers of the sixteenth and seventeenth centuries: Galileo, Newton, Leibniz, Descartes, Arnauld, Fermat, and more. Chapter three, 'The age of continuity,’ discusses eighteenth century mathematicians including Euler and Carnot, and philosophers, among them Hume, Kant and Hegel. Examining the nineteenth and early twentieth centuries, the fourth chapter describes the reduction of the continuous to the discrete, citing the contributions of Bolzano, Cauchy and Reimann. Part one of the book concludes with a chapter on divergent conceptions of the continuum, with the work of nineteenth and early twentieth century philosophers and mathematicians, including Veronese, Poincaré, Brouwer, and Weyl. Part two of this book covers contemporary mathematics, discussing topology and manifolds, categories, and functors, Grothendieck topologies, sheaves, and elementary topoi. Among the theories presented in detail are non-standard analysis, constructive and intuitionist analysis, and smooth infinitesimal analysis/synthetic differential geometry. No other book so thoroughly covers the history and development of the concepts of the continuous and the infinitesimal.
Author: Vincenzo De Risi Publisher: Springer Nature ISBN: 3030255727 Category : Science Languages : en Pages : 304
Book Description
The book offers a collection of essays on various aspects of Leibniz’s scientific thought, written by historians of science and world-leading experts on Leibniz. The essays deal with a vast array of topics on the exact sciences: Leibniz’s logic, mereology, the notion of infinity and cardinality, the foundations of geometry, the theory of curves and differential geometry, and finally dynamics and general epistemology. Several chapters attempt a reading of Leibniz’s scientific works through modern mathematical tools, and compare Leibniz’s results in these fields with 19th- and 20th-Century conceptions of them. All of them have special care in framing Leibniz’s work in historical context, and sometimes offer wider historical perspectives that go much beyond Leibniz’s researches. A special emphasis is given to effective mathematical practice rather than purely epistemological thought. The book is addressed to all scholars of the exact sciences who have an interest in historical research and Leibniz in particular, and may be useful to historians of mathematics, physics, and epistemology, mathematicians with historical interests, and philosophers of science at large.
Author: Simon Duffy Publisher: A&C Black ISBN: 1441113894 Category : Philosophy Languages : en Pages : 225
Book Description
Gilles Deleuze's engagements with mathematics, replete in his work, rely upon the construction of alternative lineages in the history of mathematics, which challenge some of the self imposed limits that regulate the canonical concepts of the discipline. For Deleuze, these challenges provide an opportunity to reconfigure particular philosophical problems - for example, the problem of individuation - and to develop new concepts in response to them. The highly original research presented in this book explores the mathematical construction of Deleuze's philosophy, as well as addressing the undervalued and often neglected question of the mathematical thinkers who influenced his work. In the wake of Alain Badiou's recent and seemingly devastating attack on the way the relation between mathematics and philosophy is configured in Deleuze's work, Simon B.Duffy offers a robust defence of the structure of Deleuze's philosophy and, in particular, the adequacy of the mathematical problems used in its construction. By reconciling Badiou and Deleuze's seemingly incompatible engagements with mathematics, Duffy succeeds in presenting a solid foundation for Deleuze's philosophy, rebuffing the recent challenges against it.
Author: Norma B. Goethe Publisher: Springer ISBN: 9401796645 Category : Science Languages : en Pages : 215
Book Description
Up to now there have been scarcely any publications on Leibniz dedicated to investigating the interrelations between philosophy and mathematics in his thought. In part this is due to the previously restricted textual basis of editions such as those produced by Gerhardt. Through recent volumes of the scientific letters and mathematical papers series of the Academy Edition scholars have obtained a much richer textual basis on which to conduct their studies - material which allows readers to see interconnections between his philosophical and mathematical ideas which have not previously been manifested. The present book draws extensively from this recently published material. The contributors are among the best in their fields. Their commissioned papers cover thematically salient aspects of the various ways in which philosophy and mathematics informed each other in Leibniz's thought.
Author: Publisher: BRILL ISBN: 9004385681 Category : Philosophy Languages : en Pages : 536
Book Description
Nicholas of Cusa and Early Modern Reform sheds new light on Cusanus’ relationship to early modernity by focusing on the reform of church, the reform of theology, the reform of perspective, and the reform of method – which together aim to encompass the breadth and depth of Cusanus’ own reform initiatives. In particular, in examining the way in which he served as inspiration for a wide and diverse array of reform-minded philosophers, ecclesiastics, theologians, and lay scholars in the midst of their struggle for the renewal and restoration of the individual, society, and the world, our volume combines a focus on Cusanus as a paradigmatic thinker with a study of his concrete influence on early modern thought. This volume is aimed at scholars working in the field of late medieval and early modern philosophy, theology, and history of science. As the first Anglophone volume to explore the early modern reception of Nicholas of Cusa, this work will provide an important complement to a growing number of companions focusing on his life and thought.
Author: Ursula Goldenbaum Publisher: Walter de Gruyter ISBN: 3110211866 Category : Philosophy Languages : en Pages : 337
Book Description
The essays offer a unified and comprehensive view of 17th century mathematical and metaphysical disputes over status of infinitesimals, particularly the question whether they were real or mere fictions. Leibniz's development of the calculus and his understanding of its metaphysical foundation are taken as both a point of departure and a frame of reference for the 17th century discussions of infinitesimals, that involved Hobbes, Wallis, Newton, Bernoulli, Hermann, and Nieuwentijt. Although the calculus was undoubtedly successful in mathematical practice, it remained controversial because its procedures seemed to lack an adequate metaphysical or methodological justification. The topic is also of philosophical interest, because Leibniz freely employed the language of infinitesimal quantities in the foundations of his dynamics and theory of forces. Thus, philosophical disputes over the Leibnizian science of bodies naturally involve questions about the nature of infinitesimals. The volume also includes newly discovered Leibnizian marginalia in the mathematical writings of Hobbes.
Author: Larry M. Jorgensen Publisher: Oxford University Press ISBN: 0191023973 Category : Philosophy Languages : en Pages : 344
Book Description
Larry M. Jorgensen provides a systematic reappraisal of Leibniz's philosophy of mind, revealing the full metaphysical background that allowed Leibniz to see farther than most of his contemporaries. In recent philosophy much effort has been put into discovering a naturalized theory of mind. Leibniz's efforts to reach a similar goal three hundred years earlier offer a critical stance from which we can assess our own theories. But while the goals might be similar, the content of Leibniz's theory significantly diverges from that of today's thought. Perhaps surprisingly, Leibniz's theological commitments yielded a thoroughgoing naturalizing methodology: the properties of an object are explicable in terms of the object's nature. Larry M. Jorgensen shows how this methodology led Leibniz to a fully natural theory of mind.
Author: Stewart Shapiro Publisher: Oxford University Press, USA ISBN: 0198809646 Category : Mathematics Languages : en Pages : 593
Book Description
Mathematical and philosophical thought about continuity has changed considerably over the ages, from Aristotle's insistence that a continuum is a unified whole, to the dominant account today, that a continuum is composed of infinitely many points. This book explores the key ideas and debates concerning continuity over more than 2500 years.
Author: John L. Bell
Author: John L. Bell
Author: Vincenzo De Risi
Author: Simon Duffy
Author: Bertrand Russell
Author: Norma B. Goethe
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Author: 
Author: Ursula Goldenbaum
Author: Larry M. Jorgensen
Author: Stewart Shapiro